Subgroups — C₂×#31771417 #46825646

Order 34432.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22575499	C₂	1	1	yes	maximal, characteristic, complemented
3	#31771417	C₂	1	16	yes	maximal, direct product factor
4	C₂×#22575486	C₂	1	3	yes	maximal, complemented
5	C₂×#22575481	C₂	1	3	yes	maximal, complemented
6	C₂×#22575475	C₂	1	3	yes	maximal, complemented
7	C₂²×#16864391	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#22575500	—	2	4	no	—
10	#22575500	C₂²	1	8	yes	complemented
11	#22575499	—	2	4	no	—
12	#22575499	C₂²	1	8	yes	complemented
13	C₂×#16864403	—	2	1	no	—
14	C₂×#16864402	C₂²	1	3	yes	—
15	C₂×#16864402	C₂²	1	3	yes	complemented
16	C₂×#16864401	—	2	3	no	—
17	C₂²×Dic₅₃₈	C₂²	1	3	yes	complemented
18	C₂×Q₈×D₂₆₉	—	2	1	no	—
19	C₂²×D₁₀₇₆	C₂²	1	3	yes	—
20	C₂×D₄×D₂₆₉	—	2	3	no	—
21	#22575486	C₂²	1	24	yes	complemented
22	#22575481	C₂²	1	24	yes	complemented
23	#22575475	C₂²	1	24	yes	complemented
24	C₂×Dic₁₀₇₆	C₂²	1	3	yes	complemented
25	C₂×#16864391	C₂²	1	6	yes	complemented
26	C₂×#16864391	C₂²	1	3	yes	complemented
27	C₂×#16864391	C₂²	1	1	yes	characteristic, complemented
28	C₂×#16864391	C₂²	1	24	yes	complemented
29	C₂×D₂₁₅₂	C₂²	1	3	yes	complemented
30	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
31	#131×C₂₆₉	C₂²	1	8	yes	complemented
32	C₂×M₁₆×C₂₆₉	C₂²	1	3	yes	complemented
33	C₂²×C₈×C₂₆₉	C₂²	1	3	yes	complemented
34	C₂³×D₂₆₉	—	2	3	no	—
35	#16864403	—	2	8	no	—
36	#16864402	—	2	6	no	—
37	#16864402	—	2	6	no	—
38	#16864402	C₂³	1	12	yes	—
39	#16864402	C₂³	1	12	yes	complemented
40	#16864401	—	2	24	no	—
41	C₂×Dic₅₃₈	—	2	3	no	—
42	C₂×Dic₅₃₈	—	2	6	no	—
43	C₂×Dic₅₃₈	C₂³	1	3	yes	—
44	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₃₈	C₂³	1	12	yes	complemented
46	Q₈×D₂₆₉	—	2	8	no	—
47	C₂×D₁₀₇₆	—	2	3	no	—
48	C₂×D₁₀₇₆	—	2	6	no	—
49	C₂×D₁₀₇₆	C₂³	1	1	yes	characteristic
50	C₂×D₁₀₇₆	C₂³	1	3	yes	—
51	C₂×D₁₀₇₆	C₂³	1	12	yes	—
52	C₂×C₂²⋊D₂₆₉	—	2	3	no	—
53	C₂×C₂²⋊D₂₆₉	—	2	3	no	—
54	D₄×D₂₆₉	—	2	24	no	—
55	C₂²×Dic₂₆₉	—	2	3	no	—
56	C₂×C₄×D₂₆₉	—	2	3	no	—
57	C₂×C₄×D₂₆₉	—	2	3	no	—
58	Dic₁₀₇₆	C₂³	1	12	yes	complemented
59	#16864391	C₂³	1	12	yes	complemented
60	#16864391	C₂³	1	24	yes	complemented
61	#16864391	C₂³	1	4	yes	complemented
62	D₂₁₅₂	C₂³	1	12	yes	complemented
63	#41×C₂₆₉	D₄	1	4	yes	—
64	#41×C₂₆₉	C₂³	1	4	yes	—
65	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
66	C₂×D₄×C₂₆₉	D₄	1	3	yes	—
67	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
68	M₁₆×C₂₆₉	C₂³	1	12	yes	complemented
69	C₂×C₈×C₂₆₉	C₂³	1	3	yes	—
70	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₆₉	C₂³	1	12	yes	complemented
72	C₂²×D₂₆₉	—	2	1	no	—
73	C₂²×D₂₆₉	—	2	3	no	—
74	C₂²×D₂₆₉	—	2	12	no	—
75	C₂²×D₂₆₉	—	8	6	no	—
76	Dic₅₃₈	—	2	12	no	—
77	Dic₅₃₈	—	2	3	no	—
78	Dic₅₃₈	—	2	1	no	—
79	Dic₅₃₈	C₂⁴	1	6	yes	—
80	Dic₅₃₈	C₂⁴	1	2	yes	—
81	D₁₀₇₆	—	2	12	no	—
82	D₁₀₇₆	—	2	3	no	—
83	D₁₀₇₆	—	2	1	no	—
84	D₁₀₇₆	C₂⁴	1	2	yes	—
85	D₁₀₇₆	C₂⁴	1	6	yes	—
86	C₂²⋊D₂₆₉	—	2	12	no	—
87	C₂²⋊D₂₆₉	—	2	12	no	—
88	C₂×Dic₂₆₉	—	2	1	no	—
89	C₂×Dic₂₆₉	—	2	3	no	—
90	C₂×Dic₂₆₉	—	2	12	no	—
91	C₄×D₂₆₉	—	2	12	no	—
92	C₄×D₂₆₉	—	2	12	no	—
93	C₂³×C₂₆₉	C₂×D₄	1	3	yes	—
94	Q₈×C₂₆₉	C₂×D₄	1	4	yes	—
95	D₄×C₂₆₉	C₂×D₄	1	12	yes	—
96	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
97	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
98	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₂₆₉	C₂⁴	1	6	yes	—
100	C₈×C₂₆₉	C₂⁴	1	2	yes	—
101	C₈×C₂₆₉	C₂⁴	1	6	yes	—
102	C₂²×C₂₆₉	—	2	6	no	—
103	C₂²×C₂₆₉	C₂²×D₄	1	6	yes	—
104	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×D₂₆₉	—	2	2	no	—
106	C₂×D₂₆₉	—	2	6	no	—
107	C₂×D₂₆₉	—	4	2	no	—
108	C₂×D₂₆₉	—	4	6	no	—
109	C₂×D₂₆₉	—	8	12	no	—
110	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₂₆₉	C₂²×D₄	1	6	yes	—
112	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
113	Dic₂₆₉	—	2	2	no	—
114	Dic₂₆₉	—	2	6	no	—
115	C₂×C₂₆₉	—	2	6	no	—
116	C₂×C₂₆₉	#577	1	2	yes	complemented
117	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
118	D₂₆₉	—	4	2	no	—
119	D₂₆₉	—	4	6	no	—
120	C₂₆₉	C₂×#577	1	1	yes	characteristic, complemented, Sylow
121	C₂×#577	—	269	1	no	maximal, Sylow
122	C₂×2₋¹⁺⁴	—	269	1	no	—
123	C₂×H(5,𝔽₂)	—	269	1	no	—
124	#577	—	269	16	no	—
125	C₂×#137	—	269	3	no	—
126	C₂×AGL(1,ℤ/8)	—	269	3	no	—
127	C₂×#135	—	269	3	no	—
128	C₂²×SD₁₆	—	269	3	no	—
129	C₂×#131	D₂₆₉	1	1	yes	characteristic, complemented
130	2₋¹⁺⁴	—	269	8	no	—
131	2₋¹⁺⁴	—	538	4	no	—
132	H(5,𝔽₂)	—	269	8	no	—
133	H(5,𝔽₂)	—	538	4	no	—
134	C₂×#41	—	269	3	no	—
135	C₂×#41	—	269	3	no	—
136	C₂×#41	—	538	1	no	—
137	C₂×#41	—	538	3	no	—
138	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
139	C₂²×Q₈	—	269	3	no	—
140	C₂²×Q₈	—	538	1	no	—
141	C₂²×D₄	—	269	3	no	—
142	C₂²×D₄	—	538	3	no	—
143	#137	—	269	24	no	—
144	AGL(1,ℤ/8)	—	269	24	no	—
145	#135	—	269	24	no	—
146	C₂×Q₁₆	—	269	3	no	—
147	C₂×SD₁₆	—	269	3	no	—
148	C₂×SD₁₆	—	269	6	no	—
149	C₂×SD₁₆	—	269	1	no	—
150	C₂×SD₁₆	—	269	24	no	—
151	C₂×D₈	—	269	3	no	—
152	#131	C₂×D₂₆₉	1	8	yes	complemented
153	C₂×M₁₆	C₂×D₂₆₉	1	3	yes	complemented
154	C₂²×C₈	C₂×D₂₆₉	1	3	yes	complemented
155	C₂⁴	—	538	3	no	—
156	#41	—	269	12	no	—
157	#41	—	269	12	no	—
158	#41	—	538	8	no	—
159	#41	—	538	24	no	—
160	#41	—	538	6	no	—
161	#41	—	538	6	no	—
162	#41	D₁₀₇₆	1	4	yes	—
163	#41	C₂²×D₂₆₉	1	4	yes	—
164	C₂×Q₈	—	269	3	no	—
165	C₂×Q₈	—	269	12	no	—
166	C₂×Q₈	—	269	1	no	—
167	C₂×Q₈	—	538	8	no	—
168	C₂×Q₈	—	538	3	no	—
169	C₂×Q₈	—	538	6	no	—
170	C₂×Q₈	D₁₀₇₆	1	1	yes	characteristic
171	C₂×D₄	—	269	1	no	—
172	C₂×D₄	—	269	3	no	—
173	C₂×D₄	—	269	12	no	—
174	C₂×D₄	—	538	3	no	—
175	C₂×D₄	—	538	3	no	—
176	C₂×D₄	—	538	24	no	—
177	C₂×D₄	—	538	3	no	—
178	C₂×D₄	—	538	6	no	—
179	C₂×D₄	D₁₀₇₆	1	3	yes	—
180	C₂²×C₄	—	538	3	no	—
181	C₂²×C₄	—	538	3	no	—
182	C₂²×C₄	—	538	3	no	—
183	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
184	Q₁₆	—	269	12	no	—
185	SD₁₆	—	269	12	no	—
186	SD₁₆	—	269	24	no	—
187	SD₁₆	—	269	4	no	—
188	D₈	—	269	12	no	—
189	M₁₆	C₂²×D₂₆₉	1	12	yes	complemented
190	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
191	C₂×C₈	C₂²×D₂₆₉	1	3	yes	—
192	C₂×C₈	C₂²×D₂₆₉	1	12	yes	complemented
193	C₂³	—	538	1	no	—
194	C₂³	—	538	3	no	—
195	C₂³	—	538	12	no	—
196	C₂³	—	2152	6	no	—
197	C₂³	C₂×D₁₀₇₆	1	3	yes	—
198	Q₈	—	269	6	no	—
199	Q₈	—	269	2	no	—
200	Q₈	—	538	12	no	—
201	Q₈	—	538	3	no	—
202	Q₈	—	538	1	no	—
203	Q₈	C₂×D₁₀₇₆	1	4	yes	—
204	D₄	—	269	2	no	—
205	D₄	—	269	6	no	—
206	D₄	—	538	12	no	—
207	D₄	—	538	12	no	—
208	D₄	—	538	12	no	—
209	D₄	—	538	3	no	—
210	D₄	—	538	1	no	—
211	D₄	C₂×D₁₀₇₆	1	12	yes	—
212	C₂×C₄	—	538	12	no	—
213	C₂×C₄	—	538	1	no	—
214	C₂×C₄	—	538	3	no	—
215	C₂×C₄	—	538	12	no	—
216	C₂×C₄	—	538	12	no	—
217	C₂×C₄	C₂×D₁₀₇₆	1	6	yes	—
218	C₂×C₄	C₂×D₁₀₇₆	1	3	yes	—
219	C₂×C₄	C₂³×D₂₆₉	1	6	yes	—
220	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
221	C₈	C₂³×D₂₆₉	1	2	yes	—
222	C₈	C₂³×D₂₆₉	1	6	yes	—
223	C₂²	—	2	6	no	—
224	C₂²	—	538	2	no	—
225	C₂²	—	538	6	no	—
226	C₂²	—	1076	6	no	—
227	C₂²	—	1076	2	no	—
228	C₂²	—	2152	12	no	—
229	C₂²	C₂²×D₁₀₇₆	1	6	yes	—
230	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
231	C₄	—	538	2	no	—
232	C₄	—	538	6	no	—
233	C₄	C₂²×D₁₀₇₆	1	6	yes	—
234	C₄	C₂²×D₁₀₇₆	1	1	yes	characteristic
235	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	1076	6	no	—
238	C₂	—	1076	2	no	—
239	C₂	#31771417	1	2	yes	direct product factor
240	C₂	C₂³×D₁₀₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₇₆
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	C₂×#131×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	C₂×#131	D₂₆₉
hypocenter	C₂₆₉	C₂×#577
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#22575500	C₂	semidirect
2. C₂×#22575499	C₂	semidirect
3. #31771417	C₂	direct (N × H)
4. C₂×#22575486	C₂	semidirect
5. C₂×#22575481	C₂	semidirect
6. C₂×#22575475	C₂	semidirect
7. C₂²×#16864391	C₂	semidirect
8. C₂×#131×C₂₆₉	C₂	semidirect
10. #22575500	C₂²	semidirect
12. #22575499	C₂²	semidirect
15. C₂×#16864402	C₂²	semidirect
17. C₂²×Dic₅₃₈	C₂²	semidirect
21. #22575486	C₂²	semidirect
22. #22575481	C₂²	semidirect
23. #22575475	C₂²	semidirect
24. C₂×Dic₁₀₇₆	C₂²	semidirect
25. C₂×#16864391	C₂²	semidirect
26. C₂×#16864391	C₂²	semidirect
27. C₂×#16864391	C₂²	semidirect
28. C₂×#16864391	C₂²	semidirect
29. C₂×D₂₁₅₂	C₂²	semidirect
31. #131×C₂₆₉	C₂²	semidirect
32. C₂×M₁₆×C₂₆₉	C₂²	semidirect
33. C₂²×C₈×C₂₆₉	C₂²	semidirect
39. #16864402	C₂³	semidirect
45. C₂×Dic₅₃₈	C₂³	semidirect
58. Dic₁₀₇₆	C₂³	semidirect
59. #16864391	C₂³	semidirect
60. #16864391	C₂³	semidirect
61. #16864391	C₂³	semidirect
62. D₂₁₅₂	C₂³	semidirect
68. M₁₆×C₂₆₉	C₂³	semidirect
71. C₂×C₈×C₂₆₉	C₂³	semidirect
116. C₂×C₂₆₉	#577	semidirect
120. C₂₆₉	C₂×#577	semidirect
129. C₂×#131	D₂₆₉	semidirect
152. #131	C₂×D₂₆₉	semidirect
153. C₂×M₁₆	C₂×D₂₆₉	semidirect
154. C₂²×C₈	C₂×D₂₆₉	semidirect
189. M₁₆	C₂²×D₂₆₉	semidirect
192. C₂×C₈	C₂²×D₂₆₉	semidirect
239. C₂	#31771417	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#22575500	2
C₂×#22575499	2
#31771417	2
C₂×#22575486	2
C₂×#22575481	2
C₂×#22575475	2
C₂²×#16864391	2
C₂×#131×C₂₆₉	2
C₂×#577	269

Subgroups — C2×#31771417 #46825646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771417 #46825646